Paco uses a spinner to select a number from 1 through 5, each with equal probability. Manu uses a different spinner to select a number from 1 through 10, each with equal probability. What is the probability that the product of Manu's number and Paco's number is less than 30? Express your answer as a common fraction.
Solution: A little casework seems like the simplest approach. First, if Paco spins a 1 or 2, it does not matter what Manu spins; the product is less than 30 regardless. If Paco spins a 3, the product will be 30 or greater only if Manu spins a 10, and both of these will be true with probability $\frac{1}{5} \cdot \frac{1}{10} = \frac{1}{50}$. If Paco spins a 4, Manu's spins of 8, 9 or 10 will tip us over the 30 barrier, and this with probability $\frac{1}{5} \cdot \frac{3}{10} = \frac{3}{50}$. If Paco spins a 5, Manu will break the 30 threshold with a 6, 7, 8, 9 or 10, probabilities being $\frac{1}{5} \cdot \frac{5}{10} = \frac{1}{10}$. The total probability for these three cases is $\frac{1+3+5}{50} = \frac{9}{50}$. But, we want the probability that the product is less than 30, so we subtract our fraction from 1 and get $\boxed{\frac{41}{50}}$.